I would simplify the integrand by factoring from the numerator, then dividing out the common factors of the numerator and denominator. Or you might find it easier to just split the fraction, then reduce each term.
You can do this with the help of integration by parts.
Now after simplification:
We divide this integration in two parts. First:
Then for we use integration by parts:
Now:
So:
Check:
integration (t + 2*t^2)/(t)^(1/2) dt - Wolfram|Alpha